Modularity in terms of the stack of elliptic curves

Question

Modular forms can be viewed as something like global sections of (some tensor power of) the canonical line bundle on the stack $\mathcal M_{\text{ell}}$ of (generalized) elliptic curves. Is it possible to state the modularity theorem in this context? It feels as if a connection between modular forms and elliptic curves should have something to do with a definition of one in terms of the other, but I can't figure out how the correspondence should work.

Answer 1

Given a normalized newform $f$ of weight $2$ and level $N$, you get an elliptic curve by taking the piece of the Jacobian of $X_0(N)$ on which the Hecke algebra acts like it does on $f$. Modularity says the converse is true: given the elliptic curve $E$ there is some modular form such that $E$ arises from this construction.

The Kodaira-Spencer map in deformation theory gives an isomorphism between the second power of what the question refers to as the canonical line bundle on $X_0(N)$ with the bundle of one-forms on $X_0(N)$, which might be the missing link you are looking for.

As the comment suggests, you're better off thinking of the variety $X_0(N)$ than the moduli stack with no level structure, because you need the level structure to make the modularity theorem true. (It's very false that every elliptic curve gets a non-constant map from $X(1)$.)